Probing the nature of $Y(4260)$ and $Z_c(3900)$ in the isospin violating process $Y(4260) \to J/\psi \eta \pi^0$

The isospin violation process $Y(4260) \to J/\psi \eta \pi^{0}$ is studied assuming that $Y(4260)$ is a $D_{1} \bar{D}+c.c.$ hadronic molecule. In association with the production of the $Z_c(3900)$, which is treated as a $D \bar{D}^{*}+c.c.$ hadronic molecule, this process can help us distinguish their molecular natures from other scenarios, since the incomplete cancellation between the charged and neutral--meson loops, which are prominent in the molecular picture only, produces a peak in the $e^+e^-\to Y(4260)\to J/\psi\eta\pi^0$ cross section at the $D_{1} \bar{D}+c.c.$ threshold and a very prominent peak in the $J/\psi \eta$ invariant mass spectrum in between the $D \bar{D}^{*}+c.c.$ thresholds; the latter being much narrower than the corresponding one in the isospin conserving channel, i.e. $J/\psi \pi^+ \pi^{-}$. The partial width of $Y(4260)\to J/\psi\eta\pi^0$ is about $4 \times 10^{-4}$ of that of $Y(4260)\to J/\psi\pi^+\pi^-$. The cross section of $e^+e^-\to Y(4260)\to J/\psi\eta\pi^0$ at the $D_{1} \bar{D}+c.c.$ threshold is about $0.05 \ \mathrm{pb}$ which is much larger than that produced by the nearby resonances. These features are the direct consequences of the assumed nature of these two states which might be accessible at the high-statistics experiments such as BESIII and LHCb.